## Solved Problems on Surds

1. Simplifying surds
3. Multiplication of surds
4. Division of surds
5. Comparison of surds
6. Rationalization of surds
7. Finding square root of surds

1. Simplification of Surds:

Click the link Exponents to recap exponents.

Simplify the surds:

a. 4√81

b. 3√432

c. (256)3/4

d. (4√1296)/4√256

Solutions:

a. 4√81 =

4√(34) = (34)1/4 = 34 × ¼ = 3

b. 3√432 =

3√ (27 × 16) =

3√ (33 × 24) = 3√ (33  × 23 × 2) =

3√ (3  × 2)3 × 2) = 3√(3  × 2)3 × 2) =

(3√(3  × 2)3 )× ( 3√2 ) = ((3  × 2)3 )1/3  × 3√2 =

(3  × 2) × 3√2 = 6 3√2

c. (256)3/4 =

(44)3/4 = 44 × ¾ = 43 = 64

d. ( 4√196)/ (4√256)

4√(16 × 81 ) = (24  × 34 )1/4  = 24 × ¼ × 34 × ¼  = 2 × 3 = 6, and

4√256) = 4√(4)4 = (44)1/4  = 4, so,

( 4√1296)/ (4√256) = 6/4 =3/2

Only like surds can be added.

Example1:

2√3 + 4√3 = 6√3

√2 and √3 cannot be added, because they are unlike surds.

Example 2:

Find the sum of √24 and √54

Solution:

Convert the given two surds into like surds as below:

√24 = √(4 × 6) = 2√6, and √54 = √(9 × 6) = 3√6

Therefore, √24 + √54 = 2√6 + 3√6 = 5√6

3. Multiplying Surds:

Example 1:

Find √12 × √48.

Solution:

√12 = √ (4 × 3) = 2 √3, and

√48 = √16 × 3 = 4√3, so,

√12 × √48 = 2 √3 × 4√3 = 8 × (√3)2 = 8 × 3 =24

Or more easily,

√12 × √48 = √12 × √(12 × 4) = √12 × √12 × √4 = 12 × 2 = 24

Example 2:

Find 51/4 × (125)0.25

Solution:

0.25 = 25/100 = 1/4

(125)0.25 = (53)1/4 = 53/4

So, 51/4 × (125)0.25 = 51/4 × (5)3/4 = 51/4 + 3/4 = 5

3. Division of Surds:

Example 1:

Divide 10 × 21/3  by 5 × 2 -2/3

Solution:

(10 × 21/3)/ (5 × 2-2/3) = 2 × 21/3 + 2/3 = 2 × 2 = 4

4. Comparison of Surds:

To compare surds, convert them into same order.

Example 1:

Which is greater: 3√4 or 4√7?

Solution:

First of all, 3√4 = 41/3 , and 4√7 = 71/4

Now, convert both surds into same order as below:

Raise each surd to power 12, which is L.C.M. of 3 and 4, the order of the two surds.

41/3 = (41/3)12  = 44 = 256, and

71/4  = (71/4)12  = 73  = 343

Since 343 > 256, so 4√7 > 3√4

Example 2:

Arrange the following surds in ascending order:

3√9, 4√11 and 6√17

Solution:

Write the given surds as follows:

91/3, 111/4 , 171/6

The surds are of order 3, 4 and 6 respectively.

Find L.C.M. of 3, 4 and 6, which is 12.

Now, raise each surd to power 12.

(91/3)12  = 912/3  = 94 = 6561,

(111/4 )12 = 1112/4 = 113 = 1331

(171/6)12 = 1712/6 = 172 = 289

Clearly we see that 91/3 > 111/4 > 171/6

Example 3:

Which is greater: √10 – √8 or √11 – √9?

Solution:

To compare, square both the surds.

(√10 – √8)2 = 10 – 8 – 2√10 × √8 = 2 – 2√80, and

(√11 – √9) = 11 – 9 – 2 √11 × √9 = 2 – 2 √99

Now, compare 2 – 2√80 and 2 – 2√99 as below:

Assume:

2 – 2√80 > 2 – 2√99

Strike out the common number 2 .

2 – 2√80 > 2 – 2√99,

So, we have:

– 2√80 > – 2√99,

Strike out the common factor – 2, and reverse the inequality

– 2√80 > – 2√99, we get: 80 < 99 which is true.

Since 80 < 99 is true, therefore our assumption 2 – 2√80 > 2 – 2√99

is also true.

Therefore,

√10 – √8 > √11 – √9

5. Rationalization of Surds

Surds are often rationalized.

Converting surds which are irrational numbers into a rational number is called rationalization.

6. What is a Rationalizing factor?

If the product of two surds is a rational number, then each factor is a rationalizing factor of the other.

How to rationalize √2?

Multiply it to itself:

√2 × √2 = 2, a rational number.

So, √2 is called the rationalizing factor of √2.

Example 2:

What is the rationalizing factor of √2 – 1?

Solution:

Multiply √2 + 1 to √2 – 1 to make √2 – 1 a rational number.

(√2 – 1) × (√2 + 1) = 2 – 1 = 1

So, √2 + 1 is the Rationalizing factor of √2 – 1

Example 3:

Make the denominator rational in:

1/ (√2 - √3)

Solution:

To rationalize denominator √2 - √3 is to make it free of square roots.

Multiply √2 - √3 with its conjugate surd √2 + √3

(√2 - √3) × (√2 + √3) = 2 – 3 = – 1

Now, the method of Rationalization is like this:

1/ (√2 - √3) =

[1/ (√2 - √3)] × [(√2 + √3)/ (√2 + √3)] =

(√2 + √3)/ [(√2 – √3)/ (√2 + √3)] =

(√2 + √3)/(- 1) = -(√2 + √3)

The denominator is rationalized (made free of surds)

Example 4:

What is the rationalizing factor of 21/3 + 2-1/3  ?

Solution:

The rationalizing factor of a1/3 + b-1/3  is:

a2/3 + b-2/3  - ab, because

(a1/3 + b-1/3 ) × (a2/3 + b-2/3  – ab) = a – b

Therefore, the rationalizing factor of 21/3 + 2-1/3   is :

22/3 + 2-2/3 – 1

Important Formulas on Rationalization Factors of Surds:

 What is the Surd? What is its Rationalizing Factor? √a √a √a + √b √a – √b 1/√a √a 1/(√a + √b) √a – √b 1/(√a + √b) √a – √b

How to find square root of surds:

Example 1:

Let us find the square root of 5 + 2√6.

Think of two numbers a and b, such that:

a + b = 5 and a × b = 6.

They are: 2 + 3 = 5 and 2 × 3 = 6

Therefore, 5 + 2√6 = (2 + 3) + 2 × (√2) × (√3)

Now, express 2 as (√2)2 and 3 as (√3)2

Therefore,

5 + 2√6 = [(√2)2 + (√3)2] + 2 × (√2) × (√3) =

[(√2)2 + (√3)2]2

√ (5 + 2√6)2 = √[(√2)2 + (√3)2]2 = √2 + √3.